ILL 1985 Problem 23

Let $\mathbb N = {1, 2, 3, . . .}$ . For real $x, y$ , set $S(x, y) = \{s | s = [nx+y], n \in \mathbb N\}$ . Prove that if $r > 1$ is a rational number, there exist real numbers $u$ and $v$ such that $$S(r, 0) \cap S(u, v) = \emptyset, S(r, 0) \cup S(u, v) = \mathbb N.$$